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arxiv: 2206.02936 · v1 · submitted 2022-06-06 · 🧮 math.GT

Bounds for Kirby-Thompson invariants of knotted surfaces

Rom\'an Aranda , Puttipong Pongtanapaisan , Cindy Zhang This is my paper

Pith reviewed 2026-05-24 11:13 UTC · model grok-4.3

classification 🧮 math.GT
keywords Kirby-Thompson invariantsknotted surfacesbridge numberpants complexdual curve complexsurface knotsknot invariants
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The pith

Kirby-Thompson invariants of knotted surfaces admit sharp lower bounds from bridge number.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes sharp lower bounds for two versions of the Kirby-Thompson invariants that quantify the complexity of knotted surfaces. One version measures distance in the pants complex of the surface; the second, newly introduced here, measures distance in the dual curve complex instead. Both invariants are shown to be bounded below in a way that is achieved exactly for infinitely many knotted surfaces whose bridge number is at most six, allowing concrete computation of the invariant values in those cases. A reader would care because the invariants now become calculable rather than merely abstract for an infinite family of examples.

Core claim

We provide sharp lower bounds for two versions of the Kirby-Thompson invariants for knotted surfaces, one of which was originally defined by Blair, Campisi, Taylor, and Tomova. The second version introduced in this paper measures distances in the dual curve complex instead of the pants complex. We compute the exact values of both KT-invariants for infinitely many knotted surfaces with bridge number at most six.

What carries the argument

The Kirby-Thompson invariant, defined via minimal distance between distinguished pants decompositions or curves in either the pants complex or the dual curve complex of the knotted surface.

If this is right

  • The lower bounds are achieved for all the constructed examples with bridge number at most six.
  • Exact numerical values of both invariants can now be listed for an infinite collection of knotted surfaces.
  • The dual-curve-complex version supplies an independent but equally bounded measure of the same surfaces.
  • Bridge number functions as a concrete lower bound for the values of both invariants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exact values may separate knotted surfaces that share the same bridge number but are not isotopic.
  • The pattern of bounds could be tested on surfaces whose bridge number exceeds six to see whether it persists.
  • The dual curve complex definition might apply to other invariants that already use pants decompositions.
  • These computable examples provide test cases for any future relation between KT-invariants and classical surface-knot quantities such as crossing number.

Load-bearing premise

The distance functions in the pants complex and dual curve complex correctly capture the intended topological complexity of the knotted surfaces, and the bridge-number classification of the example surfaces is accurate.

What would settle it

A single knotted surface with bridge number at most six whose computed Kirby-Thompson invariant lies strictly below the stated lower bound would falsify the sharpness claim.

Figures

Figures reproduced from arXiv: 2206.02936 by Cindy Zhang, Puttipong Pongtanapaisan, Rom\'an Aranda.

Figure 1
Figure 1. Figure 1: Getting a p3n; nq-bridge trisection for TpLq. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A p6, 2q-bridge tri-plane diagram for the T 2 -spin of a 2-bridge link (or knot) L given in bridge position (left). Lemma 2.5. Let L be a nontrivial 2-bridge link. The bridge number of TpLq is six. Moreover, every minimal pb; c1, c2, c3q-bridge trisection for TpLq satisfies pb; c1, c2, c3q “ p6; 2, 2, 2q. Proof. This follows from the fact that the bridge number of the unknotted torus is equal to three and … view at source ↗
Figure 3
Figure 3. Figure 3: Defining L ˚ pT q via efficient defining pairs. The ellipses represent the disk sets. The line joining P i ij to P j ij represents a geodesic path in the pants complex. We finish this section with a result about the structure of efficient defining pairs for the unlink. It is important to mention that the proof of Lemma 3.4 appears in Lemma 5.6 of [4]. One can extract more detailed conclusions from the orig… view at source ↗
Figure 4
Figure 4. Figure 4: An example of a pants decomposition P and GpPq. The graph GpPq is a tree with vertices of degree 3 or 1. The leaves (degree 1) correspond to punctures of Σ. Two leaves having the same neighbor correspond to a 2-punctured disk component in Σ ´ P. For any two punctures p, q, denote by λp,q Ă GpPq the unique path connecting p and q. The following lemma encapsulates key properties of GpPq. Lemma 4.2. Let P P P… view at source ↗
Figure 5
Figure 5. Figure 5: Outer leaves in GpPq correspond to twice punctured disks. Cut B1 open along the disk bounded by ep,q to obtain two 3-balls with trivial trivial tangles. Let pB1 , T1 q be the pn´1q-tangle that does not contain p and q, and let Σ1 “ BB1 ´T 1 . The curves in P 1 :“ P ´ tep,qu can be taken to be inside Σ1 . Denote by w the other endpoint of the edge ep,q in GpPq and let x, y be the other two edges with one en… view at source ↗
Figure 6
Figure 6. Figure 6: The pair of pants vl . The red arcs correspond to γ X vl . Proof. Let E be the 4-holed sphere where the A-move occurs. If γ X E “ H, then γ X y “ H. If not, then by assumption γ X E is (at most) two parallel properly embedded arcs in E disjoint from x (see [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: We can think that x and γ have the same ‘slope’ in the 4-holed sphere E. Proposition 4.6. Let P1 “ γ Y f and P2 “ ψ Y f be two pants decompositions for some sets of curves γ, ψ, f. Suppose λ is a path in PpΣq connecting P1 and P2 such that each curve in γ moves exactly once. Then, |x X y| ď 2 for all x P γ and y P ψ Proof. We prove this by induction on the length of λ. The base case, |λ| “ 1, is the statem… view at source ↗
Figure 8
Figure 8. Figure 8: L ˚ -invariant of the T 2 -spin of a 2-bridge link. Each column depicts bridge positions and efficient defining pairs for the unlink Li “ Ti Y Ti`1 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: L ˚ -invariant of the T 2 -spin of a 2-bridge link. A path connecting P i ij and P i ik. Proof. By Lemma 4.7, the case when a reducing curve in γ moves cannot occur. Suppose then that γ2 moves to ψ1. In particular, γ1 and ψ1 are disjoint. Since F is connected, Lemma 3.6 of [1] states that if the reducing curves γ1 and ψ1 bound distinct number of punctures, then they intersect at least four times. Thus, γ1 … view at source ↗
Figure 10
Figure 10. Figure 10: L ˚ -invariant of the spin of the Hopf link. Each column depicts bridge positions and efficient defining pairs for the unlink Li “ Ti Y Ti`1 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: L ˚ -invariant of the spin of the Hopf link. A path of length four connecting P i ij and P i ik. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (Left) Yoshikawa’s diagram of 102, which represents the 2-twist spun trefoil. (Middle) A banded bridge splitting of 102. (Right) A tri-plane diagram of 102 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: L ˚ -invariant of the 2-twist spun trefoil. Efficient defining pairs for the unlink Ti YTi`1 [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: L ˚ -invariant of the 2-twist spun trefoil. A path connecting P i ij and P i ik. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Close look at the curves in E and E 1 . Let R be the pants decomposition containing θ that corresponds to the A-move θ ÞÑ ψ1. By Claim 1, there are curves tBlu 4 l“1 in R satisfying (0)-(5). Moreover, the boundaries of the 4-holed E 2 corresponding to this A-move are tBlu 4 l“1 . Recall that ψ1 bounds a compressing disk for Ti and that p1 „i p2. Thus, ψ1 is a curve in E 2 bounding tB1, B2u. The condition … view at source ↗
Figure 16
Figure 16. Figure 16: Close look at the curves in E 2 . Theorem 6.4. Let T be a pb; 2q-bridge trisection for an embedded surface F Ă S 4 . If T is irreducible and unstabilized, then LpT q ě 3pb ` 1q. Proof. This follows from Proposition 6.3. Computations of L-invariants Example 6.5 (Spin of (2,n)-torus links). Let n P Z with |n| ě 2. Let L be a p2, nq-torus link (or knot) and let F “ SpLq. In particular L is a 2-bridge knot wi… view at source ↗
Figure 17
Figure 17. Figure 17: L-invariant of the T 2 -spin of a p2, nq-torus link. A path connecting P i ij and P i ik. 7 Results in the relative setting We now turn to the relative setting. For more detailed definitions, readers can consult [4, 10]. Let Z be a 3-ball parametrized as D2 ˆ I. We call B`Z “ D2 ˆ t1u the positive boundary and B´Z “ D2 ˆ t´1u the negative boundary. A trivial relative tangle pZ, Tq is a 3-ball Z containing… view at source ↗
Figure 18
Figure 18. Figure 18: Cutting the bridge splitting disk along a reducing sphere, a decomposing sphere, and a [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: A b “ 2 bridge trisection for the 1-holed unknotted torus U. The second row shows efficient defining pairs for each r.s.t. tangle Li . Estimating L ˚ -invariant of a ribbon disk Let F be the standard ribbon disk for the square knot. The goal of this subsection is to estimate L ˚ pFq. Recall that L is defined for relative trisections realizing the bridge number. Proposition 7.7. The minimum number of punct… view at source ↗
Figure 20
Figure 20. Figure 20: The three spanning trivial tangles appearing in a bridge trisected slice disk for the [PITH_FULL_IMAGE:figures/full_fig_p029_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Efficient defining pairs [PITH_FULL_IMAGE:figures/full_fig_p030_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: A path connecting P 1 12 and P 1 31. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: A path connecting P 2 12 and P 2 23 [PITH_FULL_IMAGE:figures/full_fig_p031_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: A path connecting P 3 31 and P 3 23. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_24.png] view at source ↗
read the original abstract

We provide sharp lower bounds for two versions of the Kirby-Thompson invariants for knotted surfaces, one of which was originally defined by Blair, Campisi, Taylor, and Tomova. The second version introduced in this paper measures distances in the dual curve complex instead of the pants complex. We compute the exact values of both KT-invariants for infinitely many knotted surfaces with bridge number at most six.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript provides sharp lower bounds for two versions of the Kirby-Thompson invariants of knotted surfaces: the original version (using the pants complex) and a new version introduced here (using the dual curve complex). It also computes the exact values of both invariants for infinitely many knotted surfaces with bridge number at most six.

Significance. If the claimed sharpness and exact computations hold, the results would advance the study of complexity measures for knotted surfaces by supplying concrete, verifiable values on an infinite family. The introduction of the dual-curve-complex variant offers a potentially independent perspective on the same topological objects.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for recognizing the potential significance of the sharp lower bounds and exact computations for the Kirby-Thompson invariants. The recommendation is listed as uncertain, but the report contains no major comments or specific concerns. Accordingly, we have no points to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a new KT-invariant variant using the dual curve complex independently of the original pants-complex version, then derives sharp lower bounds and exact values for an infinite family via explicit constructions and distance calculations on surfaces of bridge number at most six. No self-definitional equations, fitted inputs renamed as predictions, load-bearing self-citations, or imported uniqueness theorems appear in the abstract or described claims. The central results rest on direct topological computations rather than reduction to prior fitted parameters or author-overlapping citations that would force the outcome by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

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